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Human beings are born with an innate capacity to learn languages. Yet while mathematics is the language of pattern and form, many people struggle to acquire even its basic grammar. But what if we could experience math directly – just as we experience language by speaking it? Some years ago I founded an organisation, the Institute for Figuring, dedicated to the proposition that many ideas in math and science could be approached not just through equations and formulas but through concrete, physical activities. Take fractals, mathematical structures or sets with intermediate dimensionality. Coined by mathematician Benoit B. Mandelbrot, the term comes from the Latin "fractus," meaning broken. Instead of having one, two or three dimensions, a fractal will have, say, 1.89 or 2.73 dimensions.

This sounds absurd; indeed, in the late 19th century, when mathematicians began to explore such forms, they were flabbergasted, using terms like pathological to describe them. Fractals possess the strange property of "self-similarity." Zoom in on any part of a fractal, and each small section will have the same richly complicated structure as the whole. Clouds and coastlines, with what Mandelbrot called their highly "wiggled" geometries, exhibit this self-similar scaling (at least to the degree possible in nature, for true fractals with infinite levels of patterning are mathematical ideals).

Over the past year, working with Jeannine Mosely, a software engineer, I have led a project to teach people about these concepts by having them build a giant fractal model out of 48,912 business cards (all man-made models are also approximations). The finished object, known as the Mosely Snowflake Sponge, is on display at the University of Southern California's Doheny Memorial Library. Hundreds of students across campus participated in the construction, coming from departments as diverse as fine arts, psychology, cinema studies and engineering. High school students, professors, librarians and local artists took part. Together we folded business cards into cubes and linked thousands of cubes in intricate configurations that reveal the fractal's self-similar anatomy. Visual symphonies of rings and crosses – then rings of rings, rings of crosses, and crosses of crosses – became apparent to the eye, a physical manifestation of the concept of recursion.

Construction took more than 3,000 hours, and the form we made, held together by nothing more than the folded cards – no glue or Scotch tape – stands as a sculptural monument to a once heretical abstraction. The idea of making models of fractals out of business cards was dreamed up by Mosely, a leading practitioner of mathematical origami and a specialist in curved origami, which requires a serious knowledge of differential geometry. From 1996 to 2005, Mosely spent her leisure time supervising the building of a 66,048-card model of a fractal known as the Menger Sponge, named for its discoverer, Karl Menger, the Austrian mathematician, and for its resemblance to a sea sponge. Imagine a cube riddled with hundreds of square-shaped holes – it is the three-dimensional analogue of an important mathematical object known as the Cantor Set.

After she had made the Menger Sponge, Mosely realised that it was one of a whole family of fractals. Some are trivial, others cannot be made, but one was especially interesting. She named it the Snowflake Sponge for its enigmatic sixfold symmetry. Although it involved fewer cards – roughly 49,000 – it would be a greater problem to assemble. A suite of carefully designed submodules would have to be pieced together in a precise sequence so that the project would play out algorithmically in both space and time. In 2011, when the USC Libraries asked me to curate a project to engage students creatively with math and engineering, we decided to take on the challenge.

Most images of fractals are generated by computers, whose lightning-fast processing chips never tire of repeating simple routines thousands upon thousands of times. Making fractals materially is a lot harder because, as Mosely notes, "you have to think about such things as the size of a human hand in relation to three-dimensional holes." Building this fractal – and seeing participants who never thought of themselves as mathematically inclined become adept at understanding the complex spatial relationships – got me thinking about the experimental side of mathematics. What else might be possible?

At the Institute for Figuring project space in Los Angeles, we are now using Mosely's techniques in a free-form exercise. This time, with 60,000 electrically coloured business cards, our goal is open-ended. Visitors are encouraged to explore and play. Already, Tracy Tynan, a former Hollywood costume designer, has made a model of a Peano "space-filling" curve, a line that wiggles around in a way that can fill a plane completely. David Orozco, an artist and stay-at-home father of six, has developed a new construction method that creates an elegant, boxy three-dimensional lattice. An artist, Jacob Dotson, has created a new way of linking cubes into crystalline networks of octahedrons, regular eight-sided figures. Despite these explorations, none of these people will be able to pass a university geometry exam, and there will always be vast areas of math inaccessible to nonprofessionals. Nonetheless, many important mathematical concepts can be expressed without formal symbols.

After all, nature does it without an alphabet. Everyone knows about think tanks; at the Institute for Figuring, we have set ourselves up as a "play tank." By inviting people to play with ideas, we encourage them to experience not just the beauty inherent in mathematics, but its awesome structural power.